On foliations of the projective space with a Kupka component
type: Item
instance of: scholarly article
title: On foliations of the projective space with a Kupka component
zbMATH Open document ID: 1198.32015
DOI: 10.4171/LEM/55-3-2
author: Marco Brunella
published in: L'Enseignement Mathématique. 2e Série
publication date: 2010-02-05T00:00:00Z
review text: If \({\mathcal F}\) is a singular holomorphic foliation of codimension 1 on \(\mathbb {CP}(n)\), with \(n\geq 3\), a connected component \(K\) of the set of singularities of \({\mathcal F}\) is said to be a Kupka component (of \({\mathcal F}\)) if, for any \(p\in K\), there is a neighbourhood \(U\) around \(p\) such that the restriction \({\mathcal F}_{|U}\) is generated by a 1-form \(\omega\) verifying \(\omega(p)=0\) and \(d\omega(p)\neq 0\). Such a component \(K\) is smooth and of codimension 2 in \(\mathbb{CP}(n)\), and, along it, \({\mathcal F}\) is locally the product \({\mathcal F}_0\times \mathbb D^{n-2}\), where \({\mathcal F}_0\) is a singular foliation in a disk \(\mathbb D^2\). The foliation \({\mathcal F}_0\) is called the transversal type of \(K\), and after \textit{X. Gómez-Mont} and \textit{A. Lins-Neto} [Topology 30, No. 3, 315--334 (1991; Zbl 0735.57014)] one knows that it can be represented by a linear equation \(p w dz - q z dw=0\), with \(p,q\) positive coprime integers. When \(p=q=1\), \(K\) is said to be a radial Kupka component. The foliations with a non-radial Kupka component were classified by \textit{D. Cerveau} and \textit{A. Lins-Neto} in [Astérisque. 222, 93--133 (1994; Zbl 0823.32014)] and \textit{O. Calvo-Andrade} in [Bol. Soc. Bras. Mat., Nova Sér. 30, No. 2, 183--197 (1999; Zbl 1058.32023)]. They proved that \(K\) is a complete intersection, \(K=\{F=G=0\}\), and \({\mathcal F}\) is defined by a pencil of algebraic hypersurfaces \(pGdF - qFdG =0\) where \(p,q \in \mathbb Z^+\) and \(pq\geq 2\). Recently, \textit{O. Calvo-Andrade} (unpublished) has proved that this result can be extended to the radial case: If \({\mathcal F}\) is a singular holomorphic foliation of codimension 1 on \(\mathbb{CP}(n)\), with \(n\geq 3\), with a radial Kupka component \(K\), then \(K\) is a complete intersection, \(K=\{F=G=0\}\), and \({\mathcal F}\) is defined by a pencil of algebraic hypersurfaces \(GdF -FdG =0\). The main result of the paper under rev…
reviewed by: Alvaro Bustinduy
Mathematics Subject Classification ID: 32S65
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 5666403
zbMATH Keywords: foliations with Kupka component
zbMATH Keywords: projective space
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/lem/55-3-2
OpenAlex ID: W2048567892
description: scientific article; zbMATH DE number 5666403